Theorem tgsegconeq 25381

Description: Two points that satisfy the conclusion of axtgsegcon 25363 are identical. Uniqueness portion of Theorem 2.12 of [Schwabhauser] p. 29. (Contributed by Thierry Arnoux, 23-Mar-2019.)

Hypotheses

Ref	Expression
tkgeom.p	|- P = ( Base ` G )
tkgeom.d	|- .- = ( dist ` G )
tkgeom.i	|- I = (Itv ` G )
tkgeom.g	|- ( ph -> G e. TarskiG )
tgcgrextend.a	|- ( ph -> A e. P )
tgcgrextend.b	|- ( ph -> B e. P )
tgcgrextend.c	|- ( ph -> C e. P )
tgcgrextend.d	|- ( ph -> D e. P )
tgcgrextend.e	|- ( ph -> E e. P )
tgcgrextend.f	|- ( ph -> F e. P )
tgsegconeq.1	|- ( ph -> D =/= A )
tgsegconeq.2	|- ( ph -> A e. ( D I E ) )
tgsegconeq.3	|- ( ph -> A e. ( D I F ) )
tgsegconeq.4	|- ( ph -> ( A .- E ) = ( B .- C ) )
tgsegconeq.5	|- ( ph -> ( A .- F ) = ( B .- C ) )

Assertion

Ref	Expression
tgsegconeq	|- ( ph -> E = F )

Proof of Theorem tgsegconeq

Step	Hyp	Ref	Expression
1		tkgeom.p	. 2 |- P = ( Base ` G )
2		tkgeom.d	. 2 |- .- = ( dist ` G )
3		tkgeom.i	. 2 |- I = (Itv ` G )
4		tkgeom.g	. 2 |- ( ph -> G e. TarskiG )
5		tgcgrextend.e	. 2 |- ( ph -> E e. P )
6		tgcgrextend.f	. 2 |- ( ph -> F e. P )
7		tgcgrextend.d	. . . 4 |- ( ph -> D e. P )
8		tgcgrextend.a	. . . 4 |- ( ph -> A e. P )
9		tgsegconeq.1	. . . 4 |- ( ph -> D =/= A )
10		tgsegconeq.2	. . . 4 |- ( ph -> A e. ( D I E ) )
11		eqidd 2623	. . . 4 |- ( ph -> ( D .- A ) = ( D .- A ) )
12		eqidd 2623	. . . 4 |- ( ph -> ( A .- E ) = ( A .- E ) )
13		tgsegconeq.3	. . . . 5 |- ( ph -> A e. ( D I F ) )
14		tgsegconeq.4	. . . . . 6 |- ( ph -> ( A .- E ) = ( B .- C ) )
15		tgsegconeq.5	. . . . . 6 |- ( ph -> ( A .- F ) = ( B .- C ) )
16	14, 15	eqtr4d 2659	. . . . 5 |- ( ph -> ( A .- E ) = ( A .- F ) )
17	1, 2, 3, 4, 7, 8, 5, 7, 8, 6, 10, 13, 11, 16	tgcgrextend 25380	. . . 4 |- ( ph -> ( D .- E ) = ( D .- F ) )
18	1, 2, 3, 4, 7, 8, 5, 7, 8, 5, 5, 6, 9, 10, 10, 11, 12, 17, 16	axtg5seg 25364	. . 3 |- ( ph -> ( E .- E ) = ( E .- F ) )
19	18	eqcomd 2628	. 2 |- ( ph -> ( E .- F ) = ( E .- E ) )
20	1, 2, 3, 4, 5, 6, 5, 19	axtgcgrid 25362	1 |- ( ph -> E = F )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-trkgc 25347  df-trkgcb 25349  df-trkg 25352

This theorem is referenced by:  tgbtwnouttr2  25390  tgcgrxfr  25413  tgbtwnconn1lem1  25467  hlcgreulem  25512  mirreu3  25549